The Lab

Where it comes from

• Two kinds of probability. Andrey Kolmogorov (1933) set out the rules of ordinary probability: chances are numbers from 0 to 1 that add up. Quantum physics needed something else. Max Born (1926) said a particle's chance of being somewhere is the square of a wave's size there, and John von Neumann (1932) built the maths for it.
• Prices as a diffusion. Louis Bachelier (1900) modelled prices as a random walk. Fischer Black and Myron Scholes (1973), with Robert Merton (1973), priced options on that basis.
• Physics' tools. Richard Feynman (1948) wrote quantum mechanics as a sum over all paths. Mark Kac (1949) showed that the same kind of sum, with ordinary chances in place of waves, solves diffusion equations: the Feynman–Kac formula. Gian-Carlo Wick (1954) used "imaginary time" to move between the two.
• Quantum finance. Belal Baaquie (2004) wrote option and interest rate models in the language of quantum mechanics: Hamiltonians and path integrals. Emmanuel Haven and Andrei Khrennikov (2013) applied quantum probability to decisions in economics and finance.
• Quantum cognition. Jerome Busemeyer and Emmanuel Pothos, with others, used quantum probability for human choices that break ordinary probability's rules (Pothos and Busemeyer 2009; Busemeyer and Bruza 2012).

The maths, step by step

1. Black–Scholes turns into the heat equation. Write the price of the coin as S = eˣ and count time backwards from expiry, τ. Black and Scholes' equation for an option's value V becomes

∂V/∂τ = ½σ² ∂²V/∂x² + (r − ½σ²) ∂V/∂x − rV,

where σ is the volatility and r the interest rate. Put V = e^(ax + bτ) u, with a = −(r − ½σ²)/σ² and b = −r − (r − ½σ²)²/(2σ²). The extra terms cancel, and what is left is the heat equation: ∂u/∂τ = ½σ² ∂²u/∂x².

• Example: at r = 5% and σ = 50%, a = 0.3 and b = −0.06125.
• A call option at 100 on a coin at 100, at 50% volatility, for a year, with no interest: 19.74.

2. The heat equation turns into Schrödinger's, and back. The Schrödinger equation for a free particle is

iħ ∂ψ/∂t = −(ħ²/2m) ∂²ψ/∂x².

Put t = −iτ ("imaginary time", Wick's rotation). The i's cancel and it becomes ∂ψ/∂τ = (ħ/2m) ∂²ψ/∂x², the heat equation again, with ħ/m in the place of σ².

• Example: a wave e^(ikx − iωt) with ω = 2 becomes e^(ikx) e^(−2τ). It no longer swings in time. It simply fades.

So what carries over is the maths of solving these equations: kernels, sums over paths, eigenvalues. What doesn't is the waves. After the rotation, nothing swings and nothing interferes. The option price is an average of payoffs over random paths, each weighted by an ordinary chance (Feynman–Kac). The quantum version weights each path by a wave whose phases can cancel. "Quantum finance" of this kind is a change of notation. It is useful for calculation, but it predicts nothing that the classical model doesn't.

3. Where quantum probability differs: two routes. Two routes lead to the same outcome, each with a 25% chance.

• Ordinary chances add: 25% + 25% = 50%.
• Quantum amplitudes add first, then the total is squared (Born's rule). That leaves a cross term, 2 × √(0.25 × 0.25) × cos θ, where θ is how far out of step the two waves are:
  • in step (θ = 0): 100%;
  • a quarter turn (θ = 90°): 50%, as if classical;
  • opposite (θ = 180°): 0%.

4. The rule interference breaks. Ordinary probability's law of total probability:

P(A) = P(A | B) P(B) + P(A | not B) P(not B).

The quantum version adds an interference term: δ = P(A) − [P(A | B) P(B) + P(A | not B) P(not B)], with δ = 2√(P(A | B) P(B) P(A | not B) P(not B)) × cos θ.

• In people, δ isn't 0. In Amos Tversky and Eldar Shafir's (1992) gamble, as usually reported: 69% would play again after a win, 59% after a loss, but only 36% when they didn't know which.
  • The law of total probability says 0.69 × 0.5 + 0.59 × 0.5 = 64%.
  • So δ = 36% − 64% = −28 points, and a quantum model needs cos θ = −0.44.
• Those were different groups asked in different ways. Not knowing is a different situation from knowing, so nothing forces the law to hold across them.

5. Why δ is exactly 0 on one set of market data. Take 1,000 hours. In 600, funding was positive (B), and 312 of those hours closed up (A). In the other 400, 196 closed up.

• P(A) = (312 + 196) ÷ 1,000 = 0.508.
• P(A | B) P(B) + P(A | not B) P(not B) = 0.52 × 0.6 + 0.49 × 0.4 = 0.508.
• For any counts, the sum is (312/600) × (600/1000) + (196/400) × (400/1000) = (312 + 196) ÷ 1,000: the same number. δ = 0 by arithmetic. On one record of past prices, every hour is in B or not B, and the law of total probability can't fail.
• A δ that isn't 0 needs two situations that differ, such as two periods or two markets. Then it measures how they differ, and an ordinary model with changing regimes explains that without waves. The same goes for order effects (Wang and Busemeyer, 2013): reading funding before price or price before funding is the same data either way.

6. The classical explanation of fat tails. Say half of all hours are quiet (variance 1) and half are busy (variance 4), each a bell curve on its own. Mixed together, the kurtosis (0 for a bell curve) is 3 × (1² + 4²) ÷ 2 ÷ 2.5² − 3 = 1.08: fat tails, from ordinary probability. Peter Clark (1973) argued that prices run on a clock of trading, and that this mixing is where fat tails come from.

What others found

This section is other people's work, not ours.

• Quantum maths for prices works as maths. Baaquie's path integrals and Hamiltonians compute option and interest-rate prices. Their inputs are the classical model's, and so are their answers.
• Quantum probability fits some human choices. The disjunction effect (step 4) and question-order effects are fitted well by quantum models. Wang, Solloway, Shiffrin and Busemeyer (2014) confirmed in advance a prediction quantum models make about the order of questions, across many surveys. That evidence concerns people answering questions, not prices.
• Critiques. We didn't find a test where a quantum model of prices predicted better than a classical one. We didn't find a standard critique paper we could check, so we name none.

What we tested and found

This section is ours.

• The volume clock ("The volume clock: counting time in trading instead of hours"): on Bitcoin, counting time in coins traded cut the kurtosis of returns from 14.2 (hour bars) to 2.9, on every coin and in every year. Still not a bell curve. Most of the fat tail is ordinary mixing (step 6), as Clark said.
• Memory in returns ("Hidden order in crypto prices? The BDS test, before and after a volatility model"): raw daily returns were not independent (Bitcoin's score 6.58). After a standard volatility model, the score fell to 1.18, ordinary. Every 4 hours something was left (16.0), and about half of it was the week's rhythm of trading. All of it is the kind of thing classical models are built for.
• Market moods ("The touch map in calm and wild markets: does a table for each help?"): an ordinary hidden-state model, with a low-volatility and a high-volatility state, helped the touch map a little at short waits.
• The interference test wasn't run. On one record of prices, δ is 0 by arithmetic (step 5), so the test can't find anything.

In our tests, everything found had an ordinary explanation: uneven trading, volatility that clusters, and regimes. Nothing called for interference.

Credits

• A. N. Kolmogorov, Grundbegriffe der Wahrscheinlichkeitsrechnung, Springer, 1933: ordinary probability's rules.
• Max Born, "Zur Quantenmechanik der Stoßvorgänge", Zeitschrift für Physik, 1926: the squared amplitude as a chance.
• John von Neumann, Mathematische Grundlagen der Quantenmechanik, Springer, 1932: quantum probability's maths.
• Erwin Schrödinger, "Quantisierung als Eigenwertproblem", Annalen der Physik, 1926: the wave equation in step 2.
• Louis Bachelier, "Théorie de la spéculation", Annales scientifiques de l'École Normale Supérieure, 1900.
• Fischer Black and Myron Scholes, "The Pricing of Options and Corporate Liabilities", Journal of Political Economy, 1973; Robert C. Merton, "Theory of Rational Option Pricing", Bell Journal of Economics and Management Science, 1973.
• Richard P. Feynman, "Space-Time Approach to Non-Relativistic Quantum Mechanics", Reviews of Modern Physics, 1948.
• M. Kac, "On Distributions of Certain Wiener Functionals", Transactions of the American Mathematical Society, 1949: the Feynman–Kac formula.
• G. C. Wick, "Properties of Bethe-Salpeter Wave Functions", Physical Review, 1954: the rotation to imaginary time.
• Belal E. Baaquie, Quantum Finance: Path Integrals and Hamiltonians for Options and Interest Rates, Cambridge University Press, 2004.
• Emmanuel Haven and Andrei Khrennikov, Quantum Social Science, Cambridge University Press, 2013.
• Emmanuel M. Pothos and Jerome R. Busemeyer, "A Quantum Probability Explanation for Violations of 'Rational' Decision Theory", Proceedings of the Royal Society B, 2009.
• Jerome R. Busemeyer and Peter D. Bruza, Quantum Models of Cognition and Decision, Cambridge University Press, 2012.
• Amos Tversky and Eldar Shafir, "The Disjunction Effect in Choice under Uncertainty", Psychological Science, 1992: the gamble in step 4.
• Zheng Wang and Jerome R. Busemeyer, "A Quantum Question Order Model Supported by Empirical Tests of an A Priori and Precise Prediction", Topics in Cognitive Science, 2013; Zheng Wang, Tyler Solloway, Richard M. Shiffrin and Jerome R. Busemeyer, "Context Effects Produced by Question Orders Reveal Quantum Nature of Human Judgments", Proceedings of the National Academy of Sciences, 2014.
• Peter K. Clark, "A Subordinated Stochastic Process Model with Finite Variance for Speculative Prices", Econometrica, 1973.
• The worked examples, and the point that δ is 0 on one data set, are ours. That point is plain arithmetic, and others have surely made it; we don't know who first did.
• Our tests: "The volume clock: counting time in trading instead of hours", "Hidden order in crypto prices? The BDS test, before and after a volatility model" and "The touch map in calm and wild markets: does a table for each help?".
• PerpGremlin and Claude, Quantum finance: what it adds, and what it doesn't, TerminalGremlin, 2026: the tests, the code and the words.